What is compact space?

Example 6.11

The quotient $\R/\Z$ (identify $x$ and $x + n$) is homeomorphic to the circle $S^1 = \{z \in \C : \abs z = 1\}$: the map $x \mapsto \eu^{2\iu\pi x}$ passes to a continuous bijection $\R/\Z \to S^1$ (Proposition 6.10(b)); its inverse is continuous by the compactness argument of Corollary 6.14 below (Exercise 6.5 details everything, including why $\R/\Z$ is Hausdorff and compact). Likewise $[0,1]$ with endpoints glued is $S^1$, the square with opposite sides glued is the torus, and gluing is finally a theorem, not a picture.